Weird thin lines after unwrap cube uv mapping
You are running into Blender interpolating your texture a little, so that when it "smoothes" the pixels of the texture, it's grabbing the next texture over and averaging with the local color, creating an off color along your UV edges.  

You can:


Turn off interpolation, but beware, this will make the texture a little more pixely.
Unpack your UV's a little and put an "average color" in the space in between so it interpolates better.  It looks like you baked this texture, you also might want to try just re-baking after doing this.


[Edit]

Downloading and messing with the file showed that interpolation was not the issue here (or at least turning it off doesn't help).  It seems that the UV's overlap so perfectly that the line is an artifact of that.  Fixing it is simple:


Go to the UV editor, press a until all UV's are selected.
Switch pivot type to "Individual Origins"
press s and scale all the uv's down a hair, doesn't take much




Should be fixed after that.